A 'Piercing Axe' is just another axe (admitted, the naming could be confusing), so it does slashing damage = physical.
Different types of armour can offer more or less resistance against certain types of damage. E.g. Mesmer's Rogue set has +10 vs Physical, while Necromancer's Necrotic and Tormenter sets have reduced protection against Holy damage.
So, depending on who/what you are fighting certain types of damage may be more effective than others.
Weapons as such do not 'have' armour penetration, skills can add armour penetration to your attacks. E.g. Penetrating Attack (bow) or Penetrating Blow (axe).
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